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Some Curves and the Lengths of their ArcsAmelia Carolina Sparavigna

To cite this version:

Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. �hal-03236909�

Some Curves and the Lengths of their Arcs

Amelia Carolina Sparavigna

Department of Applied Science and Technology

Politecnico di Torino

Here we consider some problems from the Finkel's solution book, concerning the lengthof curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate ofBernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocalor Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on thecone and the Loxodrome. The Versiera will be discussed in detail and the link of its

name to the Versine function.

Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881

Here we consider some of the problems propose in the Finkel's solution book, havingthe full title: A mathematical solution book containing systematic solutions of many ofthe most difficult problems, Taken from the Leading Authors on Arithmetic andAlgebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus,Many Problems and Solutions from the Leading Mathematical Journals of the UnitedStates, and Many Original Problems and Solutions. with Notes and Explanations byBenjamin Franklin Finkel, 1888. A discussion on the Versiera and its history isproposed (in Italian).

Cissoid of Diocles

Conchoid of Nicomedes

Lemniscate of Bernoulli

Versiera of Agnesi and the Versine

Limaçon - Quadratrix

Spiral of Archimedes

Reciprocal or Hyperbolic spiral

Lituus

Logarithmic spiral

Curve of Pursuit

On the cone

Loxodrome

Cissoid of Diocles

The Cissoid of Diocles is the curve generated by the vertex of a parabola rolling on anequal parabola.

"A pair of parabolas face each other symmetrically:one on top and one on the bottom. Then the topparabola is rolled without slipping along the bottomone, and its successive positions are shown in theanimation. Then the path traced by the vertex of thetop parabola as it rolls is a roulette shown in red,which is cissoid of Diocles." Sam Derbyshire -en:File:RouletteAnim2.gif - "Animation of a rouletteof one parabola against another, producing a Cissoidof Diocles. Selfmade with MuPAD".

Available at en.wikipedia.org/ wiki/Cissoid_of_Diocles#/media/File: RouletteAnim2.gif

y2= x3

2a−x is the equation of the cissoid

referred do rectangular axes.

ρ=2asinθ tanθ is the polar equation of thecurve.

To find the length of an arc OAP of the cissoid.

The word "cissoid" comes from the Greek κισσοειδής kissoeidēs "ivy shaped" fromκισσός kissos "ivy" and -οειδής -oeidēs "having the likeness of". The curve is named forDiocles who studied it in the 2nd century BCE.

Conchoid of Nicomedes

This curve is described by the equation:

ρ=b sec θ ±a

The secant function (usually abbreviated as sec) is the reciprocal function of the cosinefunction.

Prob. XLV. To find the length of an arc of conchoid.

Formula s=∫√1+( drdθ )2

dθ =∫√1+ tan2θ sec2θ dθ

The length of a line given by a polar equation r(θ) can be obtained by integration. let usconsider . The length L of the line from A to B, where θ A=a , θ B=b is given

by:

Lemniscate of Bernoulli

This curve is what a Cassinian becomes when m=a . A Cassinian is a Cassini Oval,having rectangular equation:

[ y2+(a+x)2 ][ y2+(a−x)2]=m4 .

In polar coordinates: r4−2a2r2 cos2θ +a2−m4=0 . Then, the Lemniscate is:

(x+2+ y2)2=2a2(x2− y2) , r2=2a2cos 2θ

Problem. To find the length of the arc of the Lemniscate.

See details of calculation given by Makoto Kato (2012), Arc length formula for thelemniscate,https://math.stackexchange.com/q/189889.

The word Lemiscate comes from the Latin "lemniscatus" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons", or which alternatively may refer to the wool from which the ribbons were made.

The Versiera of Agnesi and the Versine

The equation of it referred to rectangular coordinates is:

x2 y=4a2(2a− y)

Let P be any point of the curve, PD= y , the ordinate of the point P and OD=x ,the abscissa. The definition of the curve requires:

EP :EF : : AO :EO or x :EF : :2a : y

EF=√(AE×EO)=√(2a− y ) y , x :√(2a− y) y : :2a: y

And then, the equation given above.

The polar equation is: r (r2−r 2sin2θ +4a2)sinθ=8a3

Problem - To find the length of an arc of the Versiera.

s=∫√1+(dydx )2

dx=∫√1+ 4 x8

(x2+4a2)4 dx

In the previous discussion, we have used the length of a chord. Here formulas from

https://mathworld.wolfram.com/CircularSegment.html

The length of the chord is:

a=2R sin(θ /2) a=2 r tan(θ /2) a=2√R2−r2 a=2√h(2 R−h)

____________________

Chi ha proposto, per la prima volta, la Versiera e come è nato il suo nome?

"II LORIA, al Cap. XII del I volume, a pp. 93-99, sotto il titolo Versiera, visiera,pseudo-versiera, scrive: « ... il nome (versiera) ora riferito s'incontra per la prima voltanelle Note al Trattato del Galileo del moto naturalmente accelerato del P. GUIDOGRANDI (1718), ove si legge che il nome versiera (latino versoria) deriva dalle parolesinus versus e che la curva fu ottenuta la prima volta dal GRANDI nell'opera intitolataQuadratura Circuli et Hyperbolae » (I ediz. 1703, II ediz. 1710). Il capitolo èinteressante e contiene anche molte indicazioni bibliografiche dalle quali risulta che lacurva venne attribuita all'AGNESI in Italia e fuori. ... Guido GRANDI, nella suaQuadratura Circuli et Hyperbolae, ... dà la sua prima definizione della versiera allaquale seguono proprietà interessanti, ne indichiamo alcune. 1) La porzione di pianocompresa fra la curva e il suo asintoto, estesa indefinitamente dalle due bande, equivalea] quadruplo del cerchio generatore della curva stessa; 2) Rotando detta porzione dipiano attorno all'asintoto si ottiene un solido equivalente al doppio di quello generatonella stessa rotazione dal cerchio generatore. Trova poi l'equazione della curva incoordinate cartesiane ortogonali, generalizzando a curve delle quali la versiera è un casoparticolare. Applica la versiera a considerazioni di ottica." ... "Ma veniamo al pregevolelavoro Instituzioni Analitiche di MARIA GAETANA AGNESI, pubblicato a Milano,nella Regia Ducal Corte nel 1748. Tratta della Versiera nel I volume in due punti: allepp. 380-81 dandone l'equazione in coordinate cartesiane xy2=a2(a−x) dove a èil diametro del cerchio generatore, e ciò era già stato fatto; alle pp. 392-93 dando unacostruzione della versiera che non è altro che la seconda costruzione del GRANDI. Enull'altro trovo su questa curva nel libro". Estratti da: Luigi Tenca, La versiera di. . .Guido Grandi. Bollettino dell’Unione Matematica Italiana, Serie 3, Vol. 12 (1957), n.3,p. 458–460. Zanichelli, link http://www.bdim.eu/item?id=BUMI_1957_3_12_3_458_0

Estratto alle Note di Grandi alle OPERE DI GALILEO GALILEIDIVISE IN QUATTRO TOMI, in questa nuova Edizione accresciute di molte cose

inedite. TOMO TERZO · Volume 3 · 1744

"Ma perché chiamarla la “versiera di Agnesi”, dal momento che la curva non è statascoperta da Gaetana Agnesi, e che lei mai ha pensato di attribuirsela? La matematicamilanese non ne porta colpa, si è limitata ad affermare – nelle sue Istituzioni analitiche,a p. 318 – che questa curva prende il nome di versiera, perché così aveva letto in unanota del matematico Guido Grandi, che aveva collaborato alla prima edizione fiorentinadelle opere del Galilei, ... Allora com’è nato l’equivoco? La spiegazione è assaisemplice se si pensa al successo dell’opera dell’Agnesi: per chi leggeva per la primavolta quel nome – e non era a conoscenza della “nota” del Grandi – la versiera era “laversiera di Agnesi”. ... E l’Agnesi scrive: «curva… che dicesi la Versiera». Tutto qui. ...Trascuriamo il fatto che in realtà l’equazione fu già studiata da Fermat sessant’anniprima e consideriamo invece quanto scrisse Guido Grandi nella sua nota ..." da Lastoria vera della “versiera” di M.G. Agnesi, https://nusquamia.wordpress.com/2019/08/05/la-storia-vera-della-versiera-di-m-g-agnesi/

In questo testo si trova una discussione del termine versiera. Dopo la discussione dellavoro di Grandi, si arriva alla distinzione tra sinus rectus e sinus versus. "Vediamoinvece che cosa sia il sinus versus. Non è un’“insenatura contraria”, o “nemica”, ma una

funzione trigonometrica, meno conosciuta delle funzioni trigonometriche abituali seno,coseno, tangente, cotangente: ma tutt’altro che un’entità misteriosa; tant’è che initaliano ha un nome, quello di “senoverso”. Viene indicato come sinv θ, o senv θ, oversen θ. Dato un angolo θ, il senoverso di θ è: senv θ = 1 – cos θ".

Il termine che appare più vicino alla "versiera", come intesa da Grandi, si ritroverebbead essere il "versine" Inglese.

"The versine or versed sine is a trigonometric function found in some of the earliest(Vedic Aryabhatia I) trigonometric tables. The versine of an angle is 1 minus its cosine.There are several related functions, most notably the coversine and haversine. Thelatter, half a versine, is of particular importance in the haversine formula of navigation."Da en.wikipedia.org/wiki/Versine

C'è anche una discussione in Inglese sulla Versiera, proposta da Evelyn Lamb, May 28,2018 in blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-witch-of-agnesi/ - Viene detto. "Grandi had given the curve the name versiera in thefirst place, saying he adapted it from the Latin word versoria, which in turn is derivedfrom the word for “turn.” In fact, in some ways 2018 is the 300th birthday not just ofAgnesi but of the witch as well; Grandi had discussed the curve in 1703, but he didn’tchristen it versiera until 1718."

Il termine versoria appare usato in navigazione come dalla "Philosphia Magnetica" diNiccolò Cabeo, 1629.

Limaçon

The name of the curve comes from the Latin limax meaning 'a snail'.

In geometry, a limaçon or limacon, also known as a limaçon of Pascal, is defined as aroulette formed by the path of a point fixed to a circle when that circle rolls around theoutside of a circle of equal radius. It can also be defined as the roulette formed when acircle rolls around a circle with half its radius so that the smaller circle is inside thelarger circle. Thus, they belong to the family of curves called centered trochoids; morespecifically, they are epitrochoids. The cardioid is the special case in which the pointgenerating the roulette lies on the rolling circle; the resulting curve has a cusp.

Construction of the limaçon with polar coordinates' origin at (x, y)=(1/2, 0) - AuthorSam Derbyshire - http://en.wikipedia.org/wiki/Image:EpitrochoidIn1.gif

Depending on the position of the point generating the curve, it may have inner and outerloops (giving the family its name), it may be heart-shaped, or it may be oval.

Equations:

(x2+ y2−ax)2=b2(x2+ y2) , r=acosθ±b with a=OA , b=PF

Quadratrix

"The quadratrix or trisectrix of Hippias (also quadratrix of Dinostratus) is a curve,which is created by a uniform motion. It is one of the oldest examples for a kinematiccurve, that is a curve created through motion. Its discovery is attributed to the Greeksophist Hippias of Elis, who used it around 420 BC in an attempt to solve the angletrisection problem (hence trisectrix). Later around 350 BC Dinostratus used it in anattempt to solve the problem of squaring the circle (hence quadratrix). ... The quadratrixis mentioned in the works of Proclus (412–485), Pappus of Alexandria (3rd and 4thcenturies) and Iamblichus (c. 240 – c. 325). Proclus names Hippias as the inventor of acurve called quadratrix and describes somewhere else how Hippias has applied thecurve on the trisection problem. Pappus only mentions how a curve named quadratrixwas used by Dinostratus, Nicomedes and others to square the circle. He neithermentions Hippias nor attributes the invention of the quadratrix to a particular person.Iamblichus just writes in a single line, that a curve called a quadratrix was used byNicomedes to square the circle. Though based on Proclus' name for the curve it isconceivable that Hippias himself used it for squaring the circle or some other curvilinearfigure, most historians of mathematics assume that Hippias invented the curve, but usedit only for the trisection of angles. Its use for squaring the circle only occurred decadeslater and was due to mathematicians like Dinostratus and Nicomedes. Thisinterpretation of the historical sources goes back to the German mathematician andhistorian Moritz Cantor." https://en.wikipedia.org/wiki/Quadratrix_of_Hippias

Spiral of Archimedes

This spiral is the locus of a point revolving about and receding from a fixed point so thatthe ratio of the radius vector to the angle through which it has moved from the polaraxis, is constant.

r=aθ

Archimedes described such a spiral in his book On Spirals. Conon of Samos was afriend of his and Pappus states that this spiral was discovered by Conon.

Spiral: From Middle French spirale, from Medieval Latin spiralis, from Latin spira, from Ancient Greek σπείρα (speíra, “wreath, coil, twist”).

Reciprocal or Hyperbolic spiral

The curve is the locus of a point revolving around and receding from a fixed point sothat it polar equation is:

r=aθ .

Because it can be generated by a circle inversion of an Archimedean spiral, it is calledreciproke spiral, too. Pierre Varignon first studied the curve in 1704. Later JohannBernoulli and Roger Cotes worked on the curve as well.

Problem: To find the length of the Hyperbolic spiral.

s=∫√r2+(d rdθ )2

dθ=a

θ 2 ∫ √(1+θ 2)dθ

s=θ √1+θ 2+log [θ +√1+θ 2]−θ−1(1+θ 2)32

s= log [θ +√1+θ 2 ]−θ−1 √1+θ 2

Lituus

In mathematics, a lituus is a spiral with polar equation: r= a√θ . Thus, the angle θ is

inversely proportional to the square of the radius r.

The curve was named for the ancient Roman lituus by Roger Cotes in a collection ofpapers entitled Harmonia Mensurarum (1722), which was published six years after hisdeath. Roger Cotes (1682 – 1716) was an English mathematician, known for workingclosely with Isaac Newton by proofreading the second edition of his famous book, thePrincipia, before publication. He also invented the quadrature formulas known asNewton–Cotes formulas, and made a geometric argument that can be interpreted as alogarithmic version of Euler's formula.

Logarithmic spiral

The Logarithmic Spiral is the locus generated by a point revolving around and recedingfrom a fixed point in such a manner that the radius vector increases in a geometricalratio, while the variable angle increases in an arithmetical ratio.

The polar equation of the logarithmic spiral: r=aθ

Problem: To find the length of the logarithmic spiral.

Formula: s=∫ √r 2+( drdθ ) dθ =∫ √r2+( r2

m2 ) dθ

m is the modulus of the system of logarithms.

"The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an"eternal line" ("ewige lini"). More than a century later, the curve was discussed byDescartes (1638), and later extensively investigated by Jacob Bernoulli, who called itSpira mirabilis, "the marvelous spiral"." en.wikipedia.org/wiki/Logarithmic_spiral

Curve of Pursuit

A fox is 80 rods north of a hound and runs directly east 350 rods before beingovertaken. How far will the hound run before catching the fox if he runs towards the foxall the time, and upon a level plain?

Let C and A be the position of the hound and fox at the start, P and mcorresponding positions of the hound and fox any time during the chase, and P' andn their positions the next instant, B the point where the hound catches the fox.CPP' B is the curve described by the hound.

Join m and P and n and P' ; they are tangents to the curve at P andP' . mo is perpendicular to P' n , and P' p is perpendicular to Pd .Pd and P' e are perpendicular to AB .

Let AC=a , AB=b , Am=x , B d= y , Pm=w , arcCP=s ,curve CPB=s1 and r the ratio of the hound's rate to the fox's rate. We have that

mn=dx , ed=P ' p=dy , PP'=ds , no−PP'=dw , s=r x (constantspeed of hound on the curve and the fox on the straight line), ds=r dx , therefore:dx /ds=1/r . From the similar right triangles P p P ' and mon :

PP ' :mn : : p P ' :no then ds :dx : :dy :no , that is: no=dx dy ds

=dy r

dy r

−ds=dw , dy−r 2dx=r dw . After integration: y−r2 x=r w+C .

When x=0 , y=0 we have that w=a , then C=−r a

y−r2 x=r w+C=r w−r a

When x=b , y=b , w=0 , we have:

b−r2b=ar , r2b−r a=b , r2−abr=1

r2−abr+ a2

4b2=a

2+4 b2

4 b2, r− a

2b= 1

2b√a2+4b2 , 2 rb−a=√a2+4b2

r b=12

(a+√a2+4 b2)=s1 since s=r x and then s1=r b .

s1=12

(a+√a2+4b2)=392.2783 rods .

This is the distance the hound runs to catch the fox. The path of the hound is known asthe "curve of pursuit". It is also know as a radiodrome.

"In geometry, a radiodrome is the pursuit curve followed by a point that is pursuinganother linearly-moving point. The term is derived from the Greek words ῥᾴδιος,rhā́idios, 'easier' and δρόμος, drómos, 'running'." en.wikipedia.org/wiki/Radiodrome

Da https://www.treccani.it/enciclopedia/curva-di-inseguimento/

Si chiama curva d’inseguimento (vedi figura), relativa alla retta r che è descritta conmoto uniforme dal punto P, la traiettoria l di un punto M che si muove con velocitàscalare costante inseguendo il punto P. L'inseguimento avviene in modo che la tangentein M alla curva l passi per il punto P. La curva d’inseguimento è detta anche curva dicaccia o curva del cane, in quanto è la traiettoria del ‘cane’ M che ‘caccia la lepre’ P (o

come nel libro di Finkel la volpe). Assunta la retta r come asse y di un riferimentocartesiano, e detto k il rapporto delle velocità scalari dei due punti M e P, l’equazionedelle curva d’inseguimento è:

y=b+ x1−k−a−2k x1+k

4 (1−k)

quando k≠1, e invece

y=b+ a ln x

2√2 ln a−x

2 √2 ln a

4a

quando k=1; le due costanti a e b sono le coordinate della posizione iniziale A del puntomobile M.

https://mathworld.wolfram.com/ApolloniusPursuitProblem.html

Apollonius Pursuit Problem: "Given a ship with a known constant direction and speedv, what course should be taken by a chase ship in pursuit (traveling at speed V) in orderto intercept the other ship in as short a time as possible? The problem can be solved byfinding all points which can be simultaneously reached by both ships, which is anApollonius circle with μ=v /V . If the circle cuts the path of the pursued ship, theintersection is the point towards which the pursuit ship should steer. If the circle doesnot cut the path, then it cannot be caught".

en.wikipedia.org/wiki/Circles_of_Apollonius

"The circles of Apollonius are any of several sets of circles associated with Apolloniusof Perga, a renowned Greek geometer. Most of these circles are found in planarEuclidean geometry, but analogs have been defined on other surfaces; for example,counterparts on the surface of a sphere can be defined through stereographicprojection".

On the cone

What is the length of a thread winding spirally round a cone, whose radius is R andaltitude a, the thread passing round n times and intersecting the slant height at equaldistance apart?

Let P be any point of the thread (x , y , z) , coordinates of the point. AnglePFC'=DOC=θ . BO=a , the altitude, DO=R the radius of the base and r

the radius of the cone at the point P. Equations of the thread are:

x=r cosθ (1)y=r sinθ (2)

z=( a2π n )θ (3)

Triangles DEF and DOB are similar:

r=Ra

(a−z)=R(1− θ2π n ) (4)

The distance between P and its consecutive position is d s=√1+(dxdz )2

+(dydz )2

dz .

s=∫d s=∫ √1+(dxdz )2

+(dydz )2

dz .

Substituting the value of r into (1) and (29 and differentiating:

dx=− R2π n [cosθ +(2π n−θ )sinθ ] dθ

dy=− R2π n [sinθ−(2π n−θ )cosθ ] dθ

dz= a2π n

dθ

Therefore:

s= ∫0

2π n adθ2π n √1+R

2

a2[cosθ +(2π n−θ )sinθ ]2+R

2

a2[sinθ −(2π n−θ )cosθ ]2

Then:

Loxodrome

The word loxodrome comes from Ancient Greek λοξός loxós: "oblique" + δρόμοςdrómos: "running" (from δραμεῖν drameîn: "to run"). The loxodrome is also known asthe rhumb line. The word rhumb may come from Spanish or Portuguese rumbo/rumo("course" or "direction") and Greek ῥόμβος rhómbos, from rhémbein.

A ship starts on the equator and travels due north-east at all time; how far has it traveledwhen its longitude, for the first time, is the same as that of the point of departure?

B is the starting point. BPN is its course. P its position at any time. N is the position atthe next instant. PN is an element of the curve, which is known as the Loxodrome orRhumb line. Let θ the longitude of the point P, ϕ the latitude. Let (x , y , z) therectangular coordinate of P. Let φ=π /4 the constant angle PNQ. r is the radiusof the sphere.

ds2=r2dϕ 2+r2cos2ϕ dθ 2

PQ=r cosϕ dθ , NQ=r dϕ , PQ /NQ= tanφ=cosϕ dθ dϕ

cosϕ dθ =tan φ dϕ

ds2=r2dϕ 2+r2 tan2φ dϕ 2 , ds=r √1+ tan2φ= rcosφ

dϕ

s= rcosφ ∫

ϕ 2

ϕ 1

dϕ= rcosφ (ϕ1−ϕ2)

We know thathttps://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions

Therefore:

θ= tanφ∫ dϕcosϕ

= tanφ ln [ tan (π /4+ϕ /2)]

eθ cotφ=tan(π /4+ϕ /2) then: ϕ=2 tan−1(eθ cotφ )−π /2

When θ=2π , ϕ=π /4 , ϕ=2 tan−1(e2π )−π /2=89 ° 47 ' 9 ' ' .6≈π /2

Loxodrome

Image Courtesy:Karl Bednarik, Peter Steinberg.